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  "body": "\n## Definition\n\nLet $X$ be a topological space, let $T \\colon X \\to X$ be a [[homeomorphism|0QT00PB5G2]], and let $r \\colon X \\to (0, \\infty)$ be a [[continuous function|0X3WGX95FE]], the *roof function*, with $\\inf_X r \u003e 0$.\nOn $X \\times \\R$, let $\\sim$ be the equivalence relation generated by\n$$(x, s + r(x)) \\sim (T(x), s) \\quad \\text{for all } x \\in X,\\ s \\in \\R.$$\nLet $X_r = (X \\times \\R) / \\sim$ with the [[quotient topology|0NQ1H2X495]], and write $[x, s]$ for the class of $(x, s)$.\nThe *suspension flow* of $T$ under $r$ is the flow $(\\varphi_t)_{t \\in \\R}$ on $X_r$ given by\n$$\\varphi_t([x, s]) = [x, s + t].$$\n\n## Remarks\n\nThe flow is well defined because the generating relation commutes with the translations $(x, s) \\mapsto (x, s + t)$.\nEvery class has a representative $(x, s)$ with $0 \\le s \u003c r(x)$.\nSo $X_r$ can be pictured as the region under the graph of $r$, with the top point $(x, r(x))$ glued to the bottom point $(T(x), 0)$.\nFor $0 \\le s \u003c r(x)$ and $t \\ge 0$, the point $\\varphi_t([x, s])$ equals $[T^n(x), s + t - \\sum_{k=0}^{n-1} r(T^k(x))]$, where $T^k$ is the $k$-th [[iterate|AYVTR2ES5B]] and $n \\ge 0$ is the number of times the orbit has hit the roof.\nThe first-return map of the flow to the base $\\set{[x, 0] : x \\in X}$ is $T$, and the return time of $[x, 0]$ is $r(x)$.\nIf $\\mu$ is a $T$-invariant measure, the product of $\\mu$ with Lebesgue measure on the region under the graph of $r$ is invariant under the flow.\n\nExample: let $X = \\R / \\Z$, $T(x) = x + \\alpha \\bmod 1$ and $r = 1$.\nThe map $[x, s] \\mapsto (x + \\alpha s, s) \\bmod \\Z^2$ is well defined, since $(x, s + 1)$ and $(x + \\alpha, s)$ both go to $(x + \\alpha s + \\alpha, s) \\bmod \\Z^2$.\nIt identifies $X_r$ with the torus $\\R^2 / \\Z^2$, and it carries the suspension flow to the linear flow $(u, v) \\mapsto (u + \\alpha t, v + t)$ in the direction $(\\alpha, 1)$.\n\nThe condition $\\inf_X r \u003e 0$ makes the sums $\\sum_{k=0}^{n-1} r(T^k(x))$ and $\\sum_{k=1}^{n} r(T^{-k}(x))$ tend to infinity, which is what gives every class a representative under the roof.\nNon-example: the doubling map $x \\mapsto 2x \\bmod 1$ is not a homeomorphism, so the construction above does not apply to it; for non-invertible maps one uses a suspension semiflow, defined only for $t \\ge 0$.\nWith $r = 1$, the space $X_r$ is the mapping torus of $T$.\n",
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